This result, in case \(w\)  is \(C^2\)  and \(A=\mathrm {Id}\) , is
[1]} (see also the previous work [2]}). As claimed there, it can be extended to more general
situations, like the one we need here. For the reader's convenience we provide a
proof (which could be generalized also to the range \(1 < \gamma \le 2\) ).

The proof is based on a Bernstein-type argument similar to that used in
Theorem IV.1 in [1]}, with some
standard modification to deal with non-divergence equations, see e.g.
[4]} or [5]}. First observe that, by a standard
bootstrap argument (triggered by Remark REF ), if \(w\in W^{2,q}_{\text{loc}}({\mathbb {R}}^N)\)  solves the above equation then \(w\)  is actually \(C^3\) .
